Electricity - Ohm's law, factors affecting resistance, and series and parallel combinations of resistors

Hello future scientists! Welcome to today's physics masterclass on one of the most fundamental chapters in Class 10 NCERT Science: Electricity.

Have you ever wondered why your smartphone charger gets slightly warm, or why plugging in a high-power geyser doesn't dim the lights in your living room? The answer lies in how electric current flows through materials and how electrical components are combined.

Grab your notebook and pen—by the end of this tutorial, you'll master Ohm's Law, understand Resistivity, and solve Series and Parallel Circuit problems with complete confidence!

1. Ohm's Law: The Heart of Circuit Theory

In 1827, German physicist Georg Simon Ohm discovered the relationship between the potential difference ($V$) applied across a conductor and the electric current ($I$) flowing through it.

Real-World Analogy: The Water Tank Model

Imagine two water tanks connected by a pipe:

• Potential Difference (Voltage, $V$): The height difference between the water level and the ground. Higher height = higher water pressure.
• Current ($I$): The rate at which water flows through the pipe.
• Resistance ($R$): The narrowness or rough texture inside the pipe that opposes water flow.

If you increase the height (voltage), water flows faster (more current). If the pipe is narrow (high resistance), water flows slower.

Statement of Ohm's Law

At a constant temperature, the electric current ($I$) flowing through a metallic conductor is directly proportional to the potential difference ($V$) applied across its ends.

$\text{Mathematically: } V \propto I$

$V = I \times R$

Where:

• $V$ = Potential Difference (measured in Volts, V)
• $I$ = Electric Current (measured in Amperes, A)
• $R$ = Resistance (measured in Ohms, $\Omega$)

What is Electric Resistance ($R$)?

Resistance is the inherent property of a conductor by which it opposes the flow of electric charges through it.

$R = \frac{V}{I}$

• 1 Ohm ($1\ \Omega$) Definition: If a potential difference of $1\text{ Volt}$ across the ends of a conductor causes a current of $1\text{ Ampere}$ to flow through it, the resistance of the conductor is said to be $1\ \Omega$.

The V-I Graph

When you plot Potential Difference ($V$) on the Y-axis against Current ($I$) on the X-axis for an ohmic conductor (like a copper wire), you get a straight line passing through the origin.

$\text{Slope of the V-I Graph} = \frac{\Delta V}{\Delta I} = \text{Resistance } (R)$

2. Factors Affecting the Resistance of a Conductor

Why do thick wires carry heavy current while thin wires are used in delicate circuits? Experiments show that the resistance $R$ of a uniform metallic conductor depends on four key factors:

1. Length of the Conductor ($l$): Resistance is directly proportional to length. A longer wire offers more collisions to moving electrons. $R \propto l$

2. Area of Cross-Section ($A$): Resistance is inversely proportional to the cross-sectional area (thickness). A thicker wire provides a wider path for electrons. $R \propto \frac{1}{A}$

3. Nature of the Material: Different materials have different internal atomic structures, offering different amounts of resistance.

4. Temperature: For pure metals, resistance increases with an increase in temperature.

Resistivity ($\rho$) – A Material Constant

Combining the physical dimensions factors:

$R \propto \frac{l}{A} \implies R = \rho \frac{l}{A}$

Where $\rho$ (rho) is a constant of proportionality called the Electrical Resistivity of the material.

$\rho = \frac{R \cdot A}{l}$

• SI Unit of Resistivity: Ohm-meter ($\Omega \cdot \text{m}$).
• Key Distinction: Resistance ($R$) depends on the length and thickness of the object, but Resistivity ($\rho$) depends ONLY on the nature of the material and temperature.
• Conductors vs. Insulators: Metals (like Copper, Aluminium) have very low resistivity ($10^{-8}$ to $10^{-6}\ \Omega \cdot \text{m}$), whereas insulators (like Rubber, Glass) have extremely high resistivity ($10^{12}$ to $10^{17}\ \Omega \cdot \text{m}$).
• Alloys: Materials like Nichrome and Constantan have higher resistivity than their constituent pure metals and do not oxidize (burn) easily at high temperatures. Hence, they are used in heating appliances like electric irons and toasters!

3. Combinations of Resistors

In practical circuits, we frequently combine two or more resistors to achieve a desired overall resistance.

A. Resistors in Series

When resistors are joined end-to-end sequentially, they are said to be connected in series.

Key Characteristics:

1. Current ($I$): The same current flows through every resistor in the series.
2. Voltage ($V$): The total voltage of the source splits across individual resistors. $V = V_1 + V_2 + V_3$

Derivation of Equivalent Resistance ($R_s$):

By Ohm's law: $V_1 = I R_1, \quad V_2 = I R_2, \quad V_3 = I R_3$

Substituting these into $V = V_1 + V_2 + V_3$: $I R_s = I R_1 + I R_2 + I R_3$

Dividing the entire equation by $I$: $R_s = R_1 + R_2 + R_3$

Takeaway: The total equivalent resistance in a series circuit is the sum of individual resistances. It is always greater than the highest individual resistance.

B. Resistors in Parallel

When resistors are connected together between two common electrical nodes, they are in a parallel combination.

Key Characteristics:

1. Voltage ($V$): The same potential difference exists across each resistor.
2. Current ($I$): The total main current divides among the branches. $I = I_1 + I_2 + I_3$

Derivation of Equivalent Resistance ($R_p$):

By Ohm's law: $I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_3 = \frac{V}{R_3}$

Substituting these into $I = I_1 + I_2 + I_3$: $\frac{V}{R_p} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$

Dividing the entire equation by $V$: $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Takeaway: The reciprocal of equivalent resistance is equal to the sum of the reciprocals of individual resistances. The overall equivalent resistance is smaller than the smallest individual resistance.

Comparison: Series vs. Parallel Circuits

4. Practice Questions with Step-by-Step Solutions

Let's test your understanding with these exam-style questions!

Question 1: Resistivity and Wire Stretching

Problem: A copper wire has a length of $2\text{ m}$ and a cross-sectional area of $1.7 \times 10^{-6}\text{ m}^2$. Its resistance is measured to be $0.02\ \Omega$.

1. Calculate the resistivity of copper.
2. What will be the new resistance if the wire's length is doubled while keeping its total volume constant (meaning its area becomes half)?

Solution:

Part 1:

• Given: $l = 2\text{ m}$, $A = 1.7 \times 10^{-6}\text{ m}^2$, $R = 0.02\ \Omega$.
• Formula: $\rho = \frac{R \cdot A}{l}$

$\rho = \frac{0.02 \times 1.7 \times 10^{-6}}{2}$ $\rho = 0.01 \times 1.7 \times 10^{-6} = 1.7 \times 10^{-8}\ \Omega \cdot \text{m}$

• Answer: The resistivity of copper is $1.7 \times 10^{-8}\ \Omega \cdot \text{m}$.

Part 2:

• New length $l' = 2l = 4\text{ m}$
• New area $A' = \frac{A}{2} = 0.85 \times 10^{-6}\text{ m}^2$
• Resistivity $\rho$ remains unchanged ($1.7 \times 10^{-8}\ \Omega \cdot \text{m}$).

$R' = \rho \frac{l'}{A'} = \rho \frac{2l}{A/2} = 4 \left(\rho \frac{l}{A}\right) = 4 R$ $R' = 4 \times 0.02\ \Omega = 0.08\ \Omega$

• Answer: The new resistance will be $0.08\ \Omega$ (it increases 4 times!).

Question 2: Parallel Resistors in a Circuit

Problem: Three resistors of $5\ \Omega$, $10\ \Omega$, and $30\ \Omega$ are connected in parallel across a $12\text{ V}$ battery. Calculate:

1. The total equivalent resistance of the circuit.
2. The total current drawn from the battery.
3. The current passing through the $10\ \Omega$ resistor.

Solution:

Part 1: Equivalent Resistance ($R_p$) $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} = \frac{1}{5} + \frac{1}{10} + \frac{1}{30}$

Taking the LCM of 5, 10, and 30 (which is 30): $\frac{1}{R_p} = \frac{6 + 3 + 1}{30} = \frac{10}{30} = \frac{1}{3}$ $R_p = 3\ \Omega$

• Answer: Total equivalent resistance = $3\ \Omega$.

Part 2: Total Circuit Current ($I_{total}$) By Ohm's Law: $I_{total} = \frac{V}{R_p} = \frac{12\text{ V}}{3\ \Omega} = 4\text{ A}$

• Answer: Total current drawn = $4\text{ Amperes}$.

Part 3: Current through $10\ \Omega$ resistor ($I_2$) In a parallel circuit, each branch gets the full battery voltage ($V = 12\text{ V}$): $I_2 = \frac{V}{R_2} = \frac{12\text{ V}}{10\ \Omega} = 1.2\text{ A}$

• Answer: Current through the $10\ \Omega$ resistor = $1.2\text{ Amperes}$.

Question 3: Mixed (Combination) Circuit Analysis

Problem: Two resistors $R_1 = 4\ \Omega$ and $R_2 = 6\ \Omega$ are connected in parallel. This combination is connected in series with a third resistor $R_3 = 3.6\ \Omega$ and a $6\text{ V}$ battery. Calculate the total circuit current.

Solution:

Step 1: Calculate the equivalent resistance of the parallel group ($R_p$) $R_p = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{4 \times 6}{4 + 6} = \frac{24}{10} = 2.4\ \Omega$

Step 2: Combine $R_p$ with the series resistor $R_3$ to find total resistance ($R_{eq}$) $R_{eq} = R_p + R_3 = 2.4\ \Omega + 3.6\ \Omega = 6.0\ \Omega$

Step 3: Apply Ohm's Law to find total current ($I$) $I = \frac{V}{R_{eq}} = \frac{6\text{ V}}{6.0\ \Omega} = 1\text{ A}$

• Answer: The total current flowing through the circuit is $1\text{ Ampere}$.

Teacher's Summary & Tips for Board Exams

1. Always write SI units in numerical answers ($V$ for Volts, $A$ for Amperes, $\Omega$ for Ohms, $\Omega\cdot\text{m}$ for Resistivity).
2. Remember that stretching or folding a wire changes its length and cross-sectional area, but its resistivity remains constant.
3. In Series, current stays constant ($I_1 = I_2 = I_{total}$).
4. In Parallel, potential difference stays constant ($V_1 = V_2 = V_{total}$).

Keep practicing circuit diagrams and numericals, and you'll easily score full marks in this chapter! Happy learning!